$Y_{\mathrm{AB}}\=\left(x\+17\right)\/6$

$Y_{\mathrm{BC}}\=\left(43-5 x\right)\/2$

$Y_{\mathrm{AC}}\=3\left(x\+1\right)\/2$

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a Triangle

Table 6.3.6(a)   Task-template implementation of the SurfaceInt command

Table 6.1.6(b)   Equations of the edges of the triangle forming region $R$ 

$F\=\left(x-3\right) \left(y-6\right)$$\stackrel{\text{assign}}{\to }$

$\sqrt{1\+{\left(\frac{\partial }{\partial x} F\right)}^2\+{\left(\frac{\partial }{\partial y} F\right)}^2}$ = $\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\lambda }$$$

Table 6.3.5(c)   Calculation of $\mathrm{\lambda }\=\sqrt{1\+F_x^2\+F_y^2}$

Iterate in the order $\mathrm{dy} \mathrm{dx}$ via the template in the Calculus palette

${\int }_1^5{\int }_{Y_{\mathrm{AB}}}^{Y_{\mathrm{AC}}}\mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_5^7{\int }_{Y_{\mathrm{AB}}}^{Y_{\mathrm{BC}}}\mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\int }_1^5{\int }_{\frac{x}{6}+\frac{17}{6}}^{\frac{3x}{2}+\frac{3}{2}}\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}\ⅆy\ⅆx+{\int }_5^7{\int }_{\frac{x}{6}+\frac{17}{6}}^{\frac{43}{2}-\frac{5x}{2}}\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}\ⅆy\ⅆx$

$\stackrel{\text{at 10 digits}}{\to }$

$39.06346474$

Table 6.2.5(d)   Iterated double-integral for finding the surface area of $F$ over $R$  

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$F≔\left(x-3\right) \left(y-6\right)\:$

Obtain the equations of the edges in region $R$ 

$$\begin{gathered} Y_{\mathrm{AB}}≔\mathrm{rhs}\left(\mathrm{isolate}\left(\mathrm{GetRepresentation}\left(\mathrm{Line}\left(\left[1\,3\right]\,\left[7\,4\right]\right)\,\mathrm{form}\=\mathrm{equation}\right)\,y\right)\right)\: \\ {Y}_{\mathrm{BC}}≔\mathrm{rhs}\left(\mathrm{isolate}\left(\mathrm{GetRepresentation}\left(\mathrm{Line}\left(\left[7\,4\right]\,\left[5\,9\right]\right)\,\mathrm{form}\=\mathrm{equation}\right)\,y\right)\right)\: \\ {Y}_{\mathrm{AC}}≔\mathrm{rhs}\left(\mathrm{isolate}\left(\mathrm{GetRepresentation}\left(\mathrm{Line}\left(\left[1\,3\right]\,\left[5\,9\right]\right)\,\mathrm{form}\=\mathrm{equation}\right)\,y\right)\right)\: \end{gathered}$$

Obtain $\mathrm{\lambda }\=\sqrt{1\+F_x^2\+F_y^2}$

$\mathrm{\λ}≔\mathrm{sqrt}\left(1\+\mathrm{diff}{\left(F\,x\right)}^2\+\mathrm{diff}{\left(F\,y\right)}^2\right)$

$\mathrm{\lambda }≔\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}$

Top-level, using the Int and evalfcommands

$$\begin{gathered} Q≔\mathrm{Int}\left(\mathrm{\λ}\,\left[y\=Y_{\mathrm{AB}}..Y_{\mathrm{AC}}\,x\=1..5\right]\right)\+\mathrm{Int}\left(\mathrm{\λ}\,\left[y\=Y_{\mathrm{AB}}..Y_{\mathrm{BC}}\,x\=5..7\right]\right)\; \\ \mathrm{evalf}\left(Q\right) \end{gathered}$$

$39.06346474$

Use the MultiInt command from the Student MultivariateCalculus package

$$\begin{gathered} \mathrm{MultiInt}\left(\mathrm{\λ}\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{AC}}\,x\=1..5\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{MultiInt}\left(\mathrm{\λ}\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{BC}}\,x\=5..7\,\mathrm{output}\=\mathrm{integral}\right) \end{gathered}$$

${\int }_1^5{\int }_{\frac{x}{6}+\frac{17}{6}}^{\frac{3x}{2}+\frac{3}{2}}\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}\ⅆy\ⅆx+{\int }_5^7{\int }_{\frac{x}{6}+\frac{17}{6}}^{\frac{43}{2}-\frac{5x}{2}}\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}\ⅆy\ⅆx$

Use the MultiInt command with a pre-defined domain option

$\mathrm{MultiInt}\left(\mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨1\,3⟩\,⟨7\,4⟩\,⟨5\,9⟩\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\int }_1^5{\int }_{\frac{3x}{2}+\frac{3}{2}}^{\frac{x}{6}+\frac{17}{6}}-\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}\ⅆy\ⅆx+{\int }_5^7{\int }_{\frac{43}{2}-\frac{5x}{2}}^{\frac{x}{6}+\frac{17}{6}}-\sqrt{1+{\left(y-6\right)}^2+{\left(x-3\right)}^2}\ⅆy\ⅆx$

$\mathrm{MultiInt}\left(\mathrm{\lambda }\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨1.\,3⟩\,⟨7.\,4⟩\,⟨5\,9⟩\right)\right)$ = $39.06346476$$$

Use the SurfaceInt command in the Student VectorCalculus package

$\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(1\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,F⟩\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨1\,3⟩\,⟨7\,4⟩\,⟨5\,9⟩\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\int }_1^5{\int }_{\frac{3x}{2}+\frac{3}{2}}^{\frac{x}{6}+\frac{17}{6}}-\sqrt{1+{\left(6-y\right)}^2+{\left(-x+3\right)}^2}\ⅆy\ⅆx+{\int }_5^7{\int }_{\frac{43}{2}-\frac{5x}{2}}^{\frac{x}{6}+\frac{17}{6}}-\sqrt{1+{\left(6-y\right)}^2+{\left(-x+3\right)}^2}\ⅆy\ⅆx$